20. Convergence of Positive Series

Homework

1. Find the sum of each geometric series or explain why it diverges.

   1. \(\displaystyle S=\sum_{n=2}^\infty \dfrac{2\cdot2^{2n}}{3^n} \)

   2. \(\displaystyle S=\sum_{n=2}^\infty \dfrac{2\cdot3^n}{2^{2n}} \)

2. Use the \(n^\text{th}\)-Term Divergence Test to determine if each series diverges or say the test is inconclusive:

   1. \(\displaystyle S=\sum_{n=1}^\infty \dfrac{n+1}{n^2} \)

   2. \(\displaystyle S=\sum_{n=1}^\infty \dfrac{n^2+1}{n^2} \)

3. Use the Integral Test to determine if each series convergence or divergence.

   1. \(\displaystyle S=\sum_{n=1}^\infty \dfrac{n+2}{(n^2+4n)^2}\)

   2. \(\displaystyle S=\sum_{n=1}^\infty \dfrac{n+2}{n^2+4n}\)

4. The series \(\displaystyle S=\sum_{n=1}^\infty e^{-n}\) converges by the integral test. If the series is approximated by \(5\) terms, the approximation is \[ S_5=\sum_{n=1}^5 e^{-n}=e^{-1}+e^{-2}+e^{-3}+e^{-4}+e^{-5} \approx0.5780553787 \] Find an upper bound on the error in this approximation. Give your answer exactly and approximately. Use a calculator to find the decimal value.
   HINT: Draw the Riemann sum rectangles which describe the error \(E_5\) and the integral region which bounds it above.

5. Determine whether each \(p\)-series converges or diverges.

   1. \(\displaystyle S=\sum_{n=1}^\infty \dfrac{\sqrt[\scriptstyle 5]{n^3}}{n}\)

   2. \(S=\dfrac{1}{\sqrt{2}\sqrt[\scriptstyle 3]{4}} +\dfrac{1}{\sqrt{3}\sqrt[\scriptstyle 3]{9}} +\dfrac{1}{\sqrt{4}\sqrt[\scriptstyle 3]{16}} +\dfrac{1}{\sqrt{5}\sqrt[\scriptstyle 3]{25}}+...\)

6. Use the Simple Comparison Test with a known series to determine the convergence or divergence of each series.

   Be careful you don't confuse a \(p\)-series, \(\displaystyle \sum_{n=1}^\infty \dfrac{1}{n^p}\), with a geometric series, \(\displaystyle \sum_{n=1}^\infty \dfrac{1}{p^n}\). In each case, \(p\) is a constant.

   1. \(\displaystyle \sum_{n=1}^\infty \dfrac{\ln n}{n^4}\)

   2. \(\displaystyle \sum_{n=1}^\infty \dfrac{\ln n}{n}\)

   3. \(\displaystyle \sum_{n=1}^\infty \dfrac{1+\sin n}{2^n}\)

7. Use the Limit Comparison Test with a known series to determine convergence or divergence of each series.

   1. \(\displaystyle \sum_{n=1}^\infty \dfrac{1+\sqrt{n^2+3n}}{1+\sqrt{n^4+n}}\)

   2. \(\displaystyle \sum_{n=1}^\infty \dfrac{2^n+3^n}{4^n}\)

   3. \(\displaystyle \sum_{n=1}^\infty \dfrac{\ln n}{n}\)

   4. \(\displaystyle \sum_{n=1}^\infty \dfrac{\ln n}{n^2}\)   (Honors Only)

8. Use the Ratio Test to determine convergence or divergence of each series.

   1. \(\displaystyle \sum_{n=1}^\infty \dfrac{2^n+3^n}{4^n}\)

   2. \(\displaystyle \sum_{n=1}^\infty \dfrac{n^3+4n+7}{n^2-3n}\)

   3. \(\displaystyle \sum_{n=1}^\infty \dfrac{n!}{e^n}\)

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Use any appropriate test to determine whether each of the following series converges or diverges. The particular test cited in the answer may not be the only appropriate test for convergence! Look back at the Overview for a summary of all the tests. It is very important you do these because they do not tell you in advance which Convergence Test to use.

9. \(\displaystyle \sum_{n=1}^\infty \dfrac{(2n)!}{(n!)^2}\)

10. \(\displaystyle \sum_{n=1}^\infty \dfrac{n}{(n^2+1)^{3/2}}\)

11. \(\displaystyle \sum_{n=1}^\infty \dfrac{n}{\sqrt{n^2+1}}\)

12. \(\displaystyle \sum_{n=1}^\infty \dfrac{1}{2^n+5^n}\)